To solve the quadratic equation \( 4x^2 + 52x + 169 = 0 \) by factoring using the perfect square trinomial pattern, we can first rewrite the equation in a form that highlights the possibility of it being a perfect square.
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Identify the coefficients: The quadratic equation can be expressed in the standard form \( ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 52 \)
- \( c = 169 \)
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Factor out the leading coefficient from the first two terms: \[ 4(x^2 + \frac{52}{4}x) + 169 = 0 \] This simplifies to: \[ 4(x^2 + 13x) + 169 = 0 \]
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Complete the square: We need to complete the square for the expression inside the parentheses. The coefficient of \( x \) is 13, so we take half of this coefficient (which is \(\frac{13}{2}\)), and then square it: \[ \left(\frac{13}{2}\right)^2 = \frac{169}{4} \] Now, we add and subtract this value inside the parentheses: \[ 4\left(x^2 + 13x + \frac{169}{4} - \frac{169}{4}\right) + 169 = 0 \] This simplifies to: \[ 4\left((x + \frac{13}{2})^2 - \frac{169}{4}\right) + 169 = 0 \]
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Distribute the 4: \[ 4(x + \frac{13}{2})^2 - 169 + 169 = 0 \] This simplifies to: \[ 4(x + \frac{13}{2})^2 = 0 \]
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Solve for \( x \): Dividing by 4: \[ (x + \frac{13}{2})^2 = 0 \] Taking the square root of both sides: \[ x + \frac{13}{2} = 0 \] Therefore: \[ x = -\frac{13}{2} \]
Conclusively, the solution to the equation \( 4x^2 + 52x + 169 = 0 \) is: \[ \boxed{-\frac{13}{2}} \]
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